Phase-hologram patterns that can shape the intensity distribution of a light beam in several planes simultaneously can be calculated with an iterative Gerchberg-Saxton algorithm [T. Haist et al., Opt. Commun. 140, 299 (1997)]. We apply this algorithm in holographic optical tweezers. This allows us to simultaneously trap several objects in individually controllable arbitrary 3-dimensional positions. We demonstrate the interactive use of our approach by trapping microscopic spheres and moving them into an arbitrary 3-dimensional configuration.
© 2004 Optical Society of America
Optical tweezers  have recently undergone a revolution made possible by the advent of commercially available spatial light modulators (SLMs) . Optical tweezers comprise a tightly focussed laser beam, which can “trap” microscopic objects at its focus. This can happen through one of various mechanisms; for example, dielectric materials (e.g. biological cells, silica spheres) experience a force towards the strong electric field at the focus where the laser is brightest. Optical tweezers have been used very widely, for example to manipulate individual biological cells  and to probe fundamental effects like the rotation of microscopic objects induced by the orbital angular momentum of the laser light .
Many applications require the simultaneous trapping of more than one object in different positions. For example, recently objects optically trapped in 2-dimensional (2D) patterns were used as a pump  for microfluidic applications. The required configurations of bright spots (foci) can be created by placing a suitable (computer-generated) hologram in the laser beam. In the form of SLMs, such holograms become reconfigurable – in our case, a computer can turn our SLM into any arbitrary phase hologram (within resolution limits). Here we describe and demonstrate experimentally the application of an algorithm by which the phase hologram pattern required to create arbitrary 3D configurations of traps can be calculated . In our implementation, the algorithm is fast enough to be used interactively.
2. Holographic optical tweezers
Optical tweezers that comprise a hologram (static or – as in the case of an SLM – reconfigurable) to shape the trapping beam are called holographic optical tweezers (HOTs). The hologram is almost always a phase hologram (the case we consider here), as intensity holograms decrease the power in the trapping beam. HOTs allow the beam to be split up into more than one trap at different positions, and to change the shape of the trap (e.g., [4, 7, 8]) or traps .
Note that there are other techniques for creating arbitrary configurations of optical traps in a plane . One widespread non-holographic technique involves periodically scanning the focus position [11, 12]; if this happens sufficiently rapidly, each of the focus positions can act as a separate optical trap. Closely related to holographic optical tweezers are tweezers that comprise a computer-controlled deformable mirror , a technology borrowed from adaptive optics. Whereas SLMs introduce a spatially dependent phase delay by effects such that passage through spatially varying liquid-crystal materials , deformable mirrors do essentially the same by altering the local path length.
HOTs first used static holograms, for example to shape the trapping beam into a doughnut mode possessing orbital angular momentum that makes trapped particles rotate ; to split the focus in arbitrary 2D patterns of foci and trap particles there ; or to shape the focus into a “light bubble” that should allow trapping of metallic particles . Computer-controlled holograms were first used in optical tweezers to trap several objects simultaneously with interactive control over the individual trap positions . The individual traps could also be shaped (for example into doughnuts), but the trapping positions were restricted to a single transverse plane, that is to 2 dimensions. The corresponding phase hologram pattern was calculated simply by superimposing in a specific way the holograms that create the individual shaped and displaced traps. This method was computationally very fast, but it also created unwanted “ghost” traps. Beam shaping that led to higher-quality traps was soon achieved by using a phase-contrast technique . As before, trapping positions were restricted to a plane, the traps could be individually shaped, and the trap positions could be controlled interactively. Interactive control over 3D trapping positions in HOTs was first achieved by generalising the algorithm used in ref.  to 3 dimensions, that is by adding lens holograms to the individual trap holograms before superimposing . This worked surprisingly well, but again suffered from ghost traps. This problem was reduced by the use of an iterative algorithm – a modification of the Gerchberg-Saxton (GS) algorithm – for the calculation of the hologram patterns . This allowed dynamic simultaneous trapping of large numbers (e.g., 200) of objects.
Here we use a different modification of the GS algorithm  to calculate the hologram patterns for the light shaping required 3D optical trapping. This GS algorithm is well suited for use in holographic optical tweezers, as it offers a combination of speed and quality: the algorithm quickly calculates a useful hologram pattern during its first iteration and refines this pattern during subsequent iterations. We demonstrate interactive simultaneous control over the 3D positions of several trapped objects.
3. Gerchberg-Saxton algorithm for intensity shaping in one plane
The GS algorithm  was originally developed to infer an electron beam’s phase distribution in a transverse plane, P, from intensity distributions in plane P and in a second plane, Q. It can also be applied to light, specifically to find a phase distribution that turns a given intensity distribution IP in plane P into a desired intensity distribution IQ in plane Q (both IP and IQ are functions of x and y). We consider here the common case of plane P being the plane of the phase hologram, IP being the intensity distribution of the illuminating laser beam, and plane Q being the far field.
Consider starting with a beam with the given intensity distribution IP and flat phase in plane P. The complex amplitude (i.e. phase and intensity) of the beam at Q, AQ, is fully determined by the complex amplitude at P, AP: AQ is simply the Fourier transform of AP. Usually the intensity distribution at Q, |AQ|2, is not the desired one, namely IQ. Naïvely, this can be “fixed” by replacing the intensity distribution at Q with IQ, while keeping the phase distribution. This change to AQ in turn affects AP – the inverse Fourier transform of AQ -, and now the intensity distribution at P is not the given one, IP, any longer. Again, we replace the “wrong” intensity with the “correct” one (in this case IP) while keeping the new phase. The process is repeated, that is the beam is calculated in alternate planes and the intensity there replaced with the correct one. The process converges, and usually within a few iterations the phase distribution at P, when imposed (with a phase hologram) on a beam with intensity IP in plane P, produces a good approximation to intensity IQ in plane Q. This GS algorithm is also known as error reduction algorithm , as each iteration decreases the errors between the desired and actual intensity distribution . Figure 1 a shows a flowchart representation of the GS algorithm.
4. … and in multiple planes
We are especially concerned with the optical-tweezers-inspired problem of generating patterns of bright spots in different planes. In the 2D algorithm described above, the phase distribution in plane Q is retained while the intensity distribution is replaced with the desired one. It is useful here to think of this step in terms of placing tiny mirrors of variable reflectivity in the positions of the spots; the reflectivity of each mirror is changed so that the power reflected from that mirror is proportional to the desired intensity of the corresponding bright spot.
This image is very easily extended to spot patterns in any number of planes: the tiny mirrors are again at the corresponding spot positions, which are now no longer all in the far-field plane but in any 3D arrangement. At the hologram, the amplitude distributions of the light reflected from each individual mirror simply add up. The remainder of the procedure remains unchanged: the resulting intensity at the hologram is replaced with that of the illuminating beam (while not altering the phase), this new beam is then again bounced off the tiny mirrors, and so on. This algorithm was first described in Ref. . Figure 1(b) shows a flow diagram of the algorithm.
The multi-plane algorithm now requires the light to be propagated not just into the far field and back, but into any arbitrary plane. This is not difficult (see, for example, ), but it is significantly more time-consuming than a single Fourier transform.
Figure 2 shows a light beam designed to have intensity cross sections in the shape of the characters “1”, “2” and “3” in different transverse planes. The intensity distributions produced by the hologram are clearly recognisable as the desired characters. However, they are not the desired uniform-intensity characters, but have instead the speckle-structure common to many beams shaped by pixellated holograms. In addition, the light forming the desired pattern in one plane is also present in all the other planes, although less intense.
This algorithm is well suited for application within interactive HOTs because it calculates useful holograms for the creation of point patterns in a single iteration: each iteration is mathematically very similar to the fast and simple lens-superposition algorithm in Ref. . However, compared to a repetition of the lens-superposition algorithm for use in dynamic HOTs, the iterative multi-plane GS algorithm offers greater flexibility (each trapping focus can, for example, be shaped into a ring by simply changing the target intensity profile accordingly) and – particularly for shaped foci – subsequent iterations improve the hologram pattern instead of calculating the same pattern repeatedly.
5. Interactive experimental demonstration in optical tweezers
To use our algorithm interactively, we developed a computer program that runs continuously through the loop shown in Fig. 1(b), while the desired intensity distributions in arbitrarily many transverse planes, as well as the number of these planes and their z positions, can be altered interactively. In our software, the intensity distributions consist of a number of points, whose number and brightness can be controlled over the keyboard and whose positions can be moved by mouse control; altering any of the trap positions (or even adding/deleting traps or trapping planes) does not reset the hologram pattern; instead the GS-algorithm loop simply keeps running with a changed target intensity pattern consisting of bright points in the new trap positions. The SLM always displays the latest phase distribution in the hologram plane. Our software does not use any further optimizations.
Figure 3 demonstrates the application in optical tweezers. It shows two examples of 3D trapping and rearrangement of multiple 2µm-diameter silica spheres. In the first example four spheres were moved out of a planar array of 13 spheres (all optically trapped at all times) and arranged into a line in a different plane. The trapping positions of the four spheres were changed interactively, one at a time. In the second example 6 spheres are manipulated individually and as two groups of 3 spheres; at one point several traps are directly above other traps without the trapped spheres being caught by the trap above or below. In that example the sequence of hologram patterns was pre-calculated. The optical tweezers arrangement was based on a NA 1.3, ×100, Zeiss Plan Neofluar oil immersion microscope objective in an inverted geometry with a Laser Quantum Excel 1500 frequency-doubled Nd:YVO4 laser, supplying a 1.5W beam at 532nm. The laser beam was expanded and reflected off a computer-controlled phase-only SLM based on a Hamamatsu PAL-SLM X7665. A 2.9GHz dual-processor computer calculated the desired holographic patterns and displayed them on the SLM. With the hologram pattern and the beam amplitude represented on a 256×256 grid, each iteration of the algorithm took ≈0.5s – fast enough for interactive use.
We have successfully applied the multi-plane GS algorithm in HOTs. This approach allows full control of the trap positions in 3D, and can run fast enough on current computers to be used interactively.
Thanks to Gabriel Spalding for very useful discussions. ZJL and JC are Royal Society University Research Fellows. We gratefully acknowledge funding from the UK Engineering and Physical Sciences Research Council (EPSRC) and from the Glasgow-Strathclyde Synergy Fund.
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